# Problems In Higher Mathematics

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Publication date
1975

Topics
mathematics, problems and solutions, algebra, calculus, geometry, series, differential equations, line integrals, double integrals, quadratic equations, solid geometry, plane geometry, ellipse, hyperbola, parabola, circle

Publisher
Mir Publishers

Collection
folkscanomy; additional_collections

Contributor
Mirtitles

Language
English

About the book:

The list of topics covered is quite exhaustive and the book has over 2500 problems and solutions. The topics covered are plane and solid analytic geometry, vector algebra, analysis, derivatives, integrals, series, differential equations etc. A good reference for those looking for many problems to solve.

The book was translated from the Russian by Yuri Ermolyev and was first published by Mir Publishers in 1975.

Chapter I.

Plane Analytic Geometry 11

1.1. Coordinates of a Point onĀ· a Straight Line and in a Plane. The Distance Between Two Points 11

1.2. Dividing a Line Segment in a Given Ratio. The Area of a Triangle and a Polygon 13

1.3. The Equation of a Line as a Locus of Points 15

1.4. The Equation of a Straight Line: (1) Slope-Intercept Form, (2) General Form, (3) Intercept Form 17

1.5. The Angle Between Two Straight Lines. The Equation of a Pencil of Straight Lines Passing Through a Given Point. The Equation of a Straight Line Passing Through Two Given Points. The Point of Intersection of Two Straight Lines 20

1.6. The Normal Equation of a Straight Line. The Distance of a Point from a Straight Line. Equations of Bisectors. The Equations of a Pencil of Straight Lines Passing Through the Point of Intersection of Two Given Straight Lines 24

1.7. Miscellaneous Problems 26

1.8. The Circle 28

1.9. The Ellipse 30

1.10. The Hyperbola 33

1.11. The Parabola 37

1.12. Directrices, Diameters, and Tangents to Curves of the Second Order 41

1.13. Transformation of Cartesian Coordinates 44

1.14. Miscellaneous Problems on Second-Order Curves 49

1.15. General Equation of a Second-Order Curve 51

1.16. Polar Coordinates 57

1.17. Algebraic Curves of the Third and Higher Orders 61

1.18. Transcendental Curves 63

Chapter 2.

Vector Algebra 64

2.1. Addition of Vectors. Multiplication of a Vector by a Scalar 64

2.2. Rectangular Coordinates of a Point and a Vector in Space 68

2.3. Scalar Product of Two Vectors 71

2.4. Vector Product of Two Vectors 75

2.5. Scalar Triple Product 78

Chapter 3.

Solid Analytic Geometry 81

3.1. The Equation of a Plane 81

3.2. Basic Problems Involving the Equation of a Plane. 83

3.3. Equations of a Straight Line in Space 86

3.4. A Straight Line and a Plane 89

3.5. Spherical and Cylindrical Surfaces 92

3.6. Conical Surfaces and Surfaces of Revolution 95

3.7. The Ellipsoid, Hyperboloids, and Paraboloids 97

Chapter 4.

Higher Algebra 101

4.1. Determinants 101

4.2. Systems of First-Degree Equations 104

4.3. Complex Numbers 108

4.4. Higher-Degree Equations. Approximate Solution of Equations 111

Chapter 5.

Introduction to Mathematical Analysis 116

5.1. Variable Quantities and Functions 116

5.2. Number Sequences. Infinitesimals and Infinities. The Limit of a Variable. The Limit of a Function 120

5.3. Basic Properties of Limits. Evaluating the Indeterminate Forms 0/0 \infty/ infty 126

5.4. The Limit of the Ratio sin(x)/x as x--> \infty a 128

5.5. Indeterminate Expressions of the Form \infty --> \infty 129

5.6. Miscellaneous Problems on Limits 129

5.7. Comparison of Infinitesimals 130

5.8. The Continuity of a Function 132

5.9. Asymptotes 136

5.10. The Number e 137

Chapter 6.

The Derivative and the Differential 139

6.1. The Derivatives of Algebraic and Trigonometric Functions 139

6.2. The Derivative of a Composite Function 141

6.3. The Tangent Line and the Normal to a Plane Curve 142

6.4. Cases of Non-differentiability of a Continuous Function 145

6.5. The Derivatives of Logarithmic and Exponential Functions 147

6.6. The Derivatives of Inverse Trigonometric Functions 149

6.7. The Derivatives of Hyperbolic Functions 150

6.8. Miscellaneous Problems on Differentiation 151

6.9. Higher-Order Derivatives 151

6.10. The Derivative of an Implicit Function 154

6.11. The Differential of a Function 156

6.12. Parametric Equations of a Curve 158

Chapter 7.

Applications of the Derivative 161

7.1. Velocity and Acceleration 161

7.2. Mean-Value Theorems 163

7.3. Evaluating Indeterminate Forms. L'Hospital's Rule 166

7.4. Increase and Decrease of a Function. Maxima and Minima 168

7.5. Finding Greatest and Least Values of a Function 172

7.6. Direction of Convexity and Points of Inflection of a Curve. Construction of Graphs 174

Chapter 8.

The Indefinite Integral 177

8.1. Indefinite Integral. Integration by Expansion 177

8.2. Integration by Substitution and Direct Integration 179

8.3. Integrals of the form dx and Those Reduced to Them 181

8.4. Integration by Parts 183

8.5. Integration of Some Trigonometric Functions 184

8.6. Integration of Rational Algebraic Functions 186

8.7. Integration of Certain Irrational Algebraic Functions 188

8.8. Integration of Certain Transcendental Functions 190

8.9. Integration of Hyperbolic Functions. Hyperbolic Substitutions 192

8.10. Miscellaneous Problems on Integration 193

Chapter 9.

The Definite Integral 195

9.1. Computing the Definite Integral 195

9.2. Computing Areas 199

9.3. The Volume of a Solid of Revolution 201

9.4. The Arc Length of a Plane Curve 203

9.5. The Area of a Surface of Revolution 205

9.6. Problems in Physics 206

9.7. ImproperIntegrals 209

9.8. The Mean Value of a Function 212

9.9. Trapezoid Rule and Simpson's Formula 213

Chapter 10.

Curvature of Plane and Space Curves 216

10.1. Curvature of a Plane Curve. The Centre and Radius of Curvature. The Evolute of a Plane Curve 216

10.2.The Arc Length of a Space Curve 218

10.3. The Derivative of a Vector Function of a Scalar Argument and Its Mechanical and Geometrical Interpretations. The Natural Trihedron of a Curve 218

10.4. Curvature and Torsion of a Space Curve 222

Chapter 11.

Partial Derivatives, Total Differentials, and Their Applications 224

11.1. Functions of Two Variables and Their Geometrical Representation 224

11.2. Partial Derivatives of the First Order 227

11.3. Total Differential of the First Order22 8

11.4. The Derivative of a Composite Function 230

11.5. Derivatives of Implicit Functions 232

11.6. Higher-Order Partial Derivatives and Total Differentials 234

11.7. Integration of Total Differentials 237

11.8. Singular Points of a Plane Curve 239

11.9. The Envelope of a Family of Plane Curves 240

11.10. The Tangent Plane and the Normal to a Surface 241

11.11. Scalar Field. Level Lines and Level Surfaces. A Derivative Along a Given Direction. Gradient 243

11.12. The Extremum of a Function of Two Variables 245

Chapter 12.

Differential Equations 248

12.1. Fundamentals 248

12.2. First-Order Differential Equation with Variables Separable. Orthogonal Trajectories 250

12.3. First-Order Differential Equations: (I) Homogeneous, (2) Linear, (3) Bernoulli's 253

12.4. Differential Equations Containing Differentials of a Product or a Quotient 255

12.5. First-Order Differential Equations in Total Differentials. Integrating Factor 255

12.6. First-Order Differential Equations Not Solved for the Derivative. Lagrange's and Clairaut's Equations 257

12.7. Differential Equations of Higher Orders Allowing for Reduction of the Order 259

12.8. Linear Homogeneous Differential Equations with Cons- tant Coefficients 261

12.9. Linear Non-homogeneous Differential Equations with Constant Coefficients 262

12.10. Differential Equations of Various Types 265

12.11. Euler's Linear Differential Equation 266

12.12. Systems of Linear Differential Equations with Constant Coefficients 266

12.13. Partial Differential Equations of the Second Order (the Method of Characteristics) 267

Chapter 13.

Double, Triple, and Line Integrals 269

13.1. Computing Areas by Means of Double Integrals 269

13.2. The Centre of Gravity and the Moment of Inertia of an Area with Uniformly Distributed Mass (for Density \mu = 1) 271

13.3. Computing Volumes by Means of Double Integrals 273

13.4. Areas of Curved Surfaces 274

13.5. The Triple Integral and Its Applications 275

13.6. The Line Integral. Green's Formula 277

13.7. Surface Integrals. Ostrogradsky's and Stokes' Formulas 281

Chapter 14.

Series 285

14.1. Numerical Series 285

14.2. Uniform Convergence of a Functional Series 288

14.3. Power Series 290

14.4. Taylor's and Maclaurin's Series 292

14.5. The Use of Series for Approximate Calculations 295

14.6. Taylor's Series for a Function of Two Variables 298

14.7. Fourier Series. Fourier Integral 299

Answers 305

Appendices 383

The list of topics covered is quite exhaustive and the book has over 2500 problems and solutions. The topics covered are plane and solid analytic geometry, vector algebra, analysis, derivatives, integrals, series, differential equations etc. A good reference for those looking for many problems to solve.

The book was translated from the Russian by Yuri Ermolyev and was first published by Mir Publishers in 1975.

Chapter I.

Plane Analytic Geometry 11

1.1. Coordinates of a Point onĀ· a Straight Line and in a Plane. The Distance Between Two Points 11

1.2. Dividing a Line Segment in a Given Ratio. The Area of a Triangle and a Polygon 13

1.3. The Equation of a Line as a Locus of Points 15

1.4. The Equation of a Straight Line: (1) Slope-Intercept Form, (2) General Form, (3) Intercept Form 17

1.5. The Angle Between Two Straight Lines. The Equation of a Pencil of Straight Lines Passing Through a Given Point. The Equation of a Straight Line Passing Through Two Given Points. The Point of Intersection of Two Straight Lines 20

1.6. The Normal Equation of a Straight Line. The Distance of a Point from a Straight Line. Equations of Bisectors. The Equations of a Pencil of Straight Lines Passing Through the Point of Intersection of Two Given Straight Lines 24

1.7. Miscellaneous Problems 26

1.8. The Circle 28

1.9. The Ellipse 30

1.10. The Hyperbola 33

1.11. The Parabola 37

1.12. Directrices, Diameters, and Tangents to Curves of the Second Order 41

1.13. Transformation of Cartesian Coordinates 44

1.14. Miscellaneous Problems on Second-Order Curves 49

1.15. General Equation of a Second-Order Curve 51

1.16. Polar Coordinates 57

1.17. Algebraic Curves of the Third and Higher Orders 61

1.18. Transcendental Curves 63

Chapter 2.

Vector Algebra 64

2.1. Addition of Vectors. Multiplication of a Vector by a Scalar 64

2.2. Rectangular Coordinates of a Point and a Vector in Space 68

2.3. Scalar Product of Two Vectors 71

2.4. Vector Product of Two Vectors 75

2.5. Scalar Triple Product 78

Chapter 3.

Solid Analytic Geometry 81

3.1. The Equation of a Plane 81

3.2. Basic Problems Involving the Equation of a Plane. 83

3.3. Equations of a Straight Line in Space 86

3.4. A Straight Line and a Plane 89

3.5. Spherical and Cylindrical Surfaces 92

3.6. Conical Surfaces and Surfaces of Revolution 95

3.7. The Ellipsoid, Hyperboloids, and Paraboloids 97

Chapter 4.

Higher Algebra 101

4.1. Determinants 101

4.2. Systems of First-Degree Equations 104

4.3. Complex Numbers 108

4.4. Higher-Degree Equations. Approximate Solution of Equations 111

Chapter 5.

Introduction to Mathematical Analysis 116

5.1. Variable Quantities and Functions 116

5.2. Number Sequences. Infinitesimals and Infinities. The Limit of a Variable. The Limit of a Function 120

5.3. Basic Properties of Limits. Evaluating the Indeterminate Forms 0/0 \infty/ infty 126

5.4. The Limit of the Ratio sin(x)/x as x--> \infty a 128

5.5. Indeterminate Expressions of the Form \infty --> \infty 129

5.6. Miscellaneous Problems on Limits 129

5.7. Comparison of Infinitesimals 130

5.8. The Continuity of a Function 132

5.9. Asymptotes 136

5.10. The Number e 137

Chapter 6.

The Derivative and the Differential 139

6.1. The Derivatives of Algebraic and Trigonometric Functions 139

6.2. The Derivative of a Composite Function 141

6.3. The Tangent Line and the Normal to a Plane Curve 142

6.4. Cases of Non-differentiability of a Continuous Function 145

6.5. The Derivatives of Logarithmic and Exponential Functions 147

6.6. The Derivatives of Inverse Trigonometric Functions 149

6.7. The Derivatives of Hyperbolic Functions 150

6.8. Miscellaneous Problems on Differentiation 151

6.9. Higher-Order Derivatives 151

6.10. The Derivative of an Implicit Function 154

6.11. The Differential of a Function 156

6.12. Parametric Equations of a Curve 158

Chapter 7.

Applications of the Derivative 161

7.1. Velocity and Acceleration 161

7.2. Mean-Value Theorems 163

7.3. Evaluating Indeterminate Forms. L'Hospital's Rule 166

7.4. Increase and Decrease of a Function. Maxima and Minima 168

7.5. Finding Greatest and Least Values of a Function 172

7.6. Direction of Convexity and Points of Inflection of a Curve. Construction of Graphs 174

Chapter 8.

The Indefinite Integral 177

8.1. Indefinite Integral. Integration by Expansion 177

8.2. Integration by Substitution and Direct Integration 179

8.3. Integrals of the form dx and Those Reduced to Them 181

8.4. Integration by Parts 183

8.5. Integration of Some Trigonometric Functions 184

8.6. Integration of Rational Algebraic Functions 186

8.7. Integration of Certain Irrational Algebraic Functions 188

8.8. Integration of Certain Transcendental Functions 190

8.9. Integration of Hyperbolic Functions. Hyperbolic Substitutions 192

8.10. Miscellaneous Problems on Integration 193

Chapter 9.

The Definite Integral 195

9.1. Computing the Definite Integral 195

9.2. Computing Areas 199

9.3. The Volume of a Solid of Revolution 201

9.4. The Arc Length of a Plane Curve 203

9.5. The Area of a Surface of Revolution 205

9.6. Problems in Physics 206

9.7. ImproperIntegrals 209

9.8. The Mean Value of a Function 212

9.9. Trapezoid Rule and Simpson's Formula 213

Chapter 10.

Curvature of Plane and Space Curves 216

10.1. Curvature of a Plane Curve. The Centre and Radius of Curvature. The Evolute of a Plane Curve 216

10.2.The Arc Length of a Space Curve 218

10.3. The Derivative of a Vector Function of a Scalar Argument and Its Mechanical and Geometrical Interpretations. The Natural Trihedron of a Curve 218

10.4. Curvature and Torsion of a Space Curve 222

Chapter 11.

Partial Derivatives, Total Differentials, and Their Applications 224

11.1. Functions of Two Variables and Their Geometrical Representation 224

11.2. Partial Derivatives of the First Order 227

11.3. Total Differential of the First Order22 8

11.4. The Derivative of a Composite Function 230

11.5. Derivatives of Implicit Functions 232

11.6. Higher-Order Partial Derivatives and Total Differentials 234

11.7. Integration of Total Differentials 237

11.8. Singular Points of a Plane Curve 239

11.9. The Envelope of a Family of Plane Curves 240

11.10. The Tangent Plane and the Normal to a Surface 241

11.11. Scalar Field. Level Lines and Level Surfaces. A Derivative Along a Given Direction. Gradient 243

11.12. The Extremum of a Function of Two Variables 245

Chapter 12.

Differential Equations 248

12.1. Fundamentals 248

12.2. First-Order Differential Equation with Variables Separable. Orthogonal Trajectories 250

12.3. First-Order Differential Equations: (I) Homogeneous, (2) Linear, (3) Bernoulli's 253

12.4. Differential Equations Containing Differentials of a Product or a Quotient 255

12.5. First-Order Differential Equations in Total Differentials. Integrating Factor 255

12.6. First-Order Differential Equations Not Solved for the Derivative. Lagrange's and Clairaut's Equations 257

12.7. Differential Equations of Higher Orders Allowing for Reduction of the Order 259

12.8. Linear Homogeneous Differential Equations with Cons- tant Coefficients 261

12.9. Linear Non-homogeneous Differential Equations with Constant Coefficients 262

12.10. Differential Equations of Various Types 265

12.11. Euler's Linear Differential Equation 266

12.12. Systems of Linear Differential Equations with Constant Coefficients 266

12.13. Partial Differential Equations of the Second Order (the Method of Characteristics) 267

Chapter 13.

Double, Triple, and Line Integrals 269

13.1. Computing Areas by Means of Double Integrals 269

13.2. The Centre of Gravity and the Moment of Inertia of an Area with Uniformly Distributed Mass (for Density \mu = 1) 271

13.3. Computing Volumes by Means of Double Integrals 273

13.4. Areas of Curved Surfaces 274

13.5. The Triple Integral and Its Applications 275

13.6. The Line Integral. Green's Formula 277

13.7. Surface Integrals. Ostrogradsky's and Stokes' Formulas 281

Chapter 14.

Series 285

14.1. Numerical Series 285

14.2. Uniform Convergence of a Functional Series 288

14.3. Power Series 290

14.4. Taylor's and Maclaurin's Series 292

14.5. The Use of Series for Approximate Calculations 295

14.6. Taylor's Series for a Function of Two Variables 298

14.7. Fourier Series. Fourier Integral 299

Answers 305

Appendices 383

Identifier
ProblemsInHigherMathematicsMinorsky

Identifier-ark
ark:/13960/t8tb3gm6s

Ocr
ABBYY FineReader 9.0

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600

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Internet Archive HTML5 Uploader 1.4.2

Year
1975

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